Binomial trees as dynamical systems

One of the simplest and very popular techniques for pricing an option or other derivative involves constructing what is known as a binomial tree. This is a tree which represents the possible paths, that might be followed by the underlying assets price. We will view this tree as a dynamical system, which means that we specify a space, and a map acting on it. Here the space will be the space of all possible paths, and the corresponding map will be the shift on each path. Such approach reveals a dynamical nature of certain financial terms and financial principles. For example, returns along a path could be defined by a potential, and the price on each path is expressed in a very “dynamical” fashion. Using this interpretation, we introduce a new characteristic as the pressure of the potential of returns. Under the conditions of no arbitrage, the pressure has to be equal to be the interest rate. This gives a new formulation of the no arbitrage principle: the expected price has to be finite: a smaller or greater discount would give either very small (zero at the limit), or very large (at the limit infinity) price. Therefore, the presented work links the discrete models of option pricing to the thermodynamical formalism and multifractal analysis of invariant sets in dynamical systems.